How Do I Fix This Drywall Hole?

So I have a hole in this drywall. What happened was water was leaking from the vent shown in the photos below, which probably caused some rotting of the wood, although it's hard to say exactly what happened because some type of animal was also living in that same area, you can see a few eggs in the far left corner of the photo.

So anyway, I'm trying to figure out how to fix it. My best guess is to either run a new piece of wood parallel to the damaged one, and then a couple of pieces of wood perpendicular to the one that has rotted out a bit, those would be to secure the new drywall after I cut out the section with the hole. That's what the blue lines represent in the photo. I don't know if I'd need to run a whole new piece parallel to the rotted one, or just shorter piece nailed to it, or leave it as is. I'd have to check again to see how bad the damage is, but there is some as is obvious in the photo.

What other options do I have, especially anything easier because the crawl space above the ceiling is fairly small. I guess I could also cut out a square around the hole, put a piece of wood in it going across the hole, screw it in on the sides, and put the new piece in (Shown below).

Ideally, you'd replace any of the drywall that had gotten wet, not just fix the hole. If you cut out to the center of a stud, then you can screw the new piece into the studs and probably not need any blocking.

Drywall does not have much structural integrity to start with and if you get it wet, it loses what little it had. As far as I'm concerned, if it gets wet, you replace it... It's not like you want to leave something up there that might be moldy, right? Rip out everything that has gotten wet and then keep ripping until you get to the ceiling joists so that you can secure the edges of the piece that you are going to cut for it.

It's actually easier to repair a fairly large hole than a small one. As already suggested, cut to the center of a stud on both sides making a rectangular hole. It's easier to cut a matching rectangle that an odd shape. I have had occasions when I sistered a piece of 2x4 on to the stud(s) to give more latitude in nailing the patch. What hasn't been mentioned is to be certain the leak that caused the damage has been repaired before you patch.

On small sheetrock patches, I'll cut the piece so that it is a few inches larger than the original hole and then cut everything away on the overlap except the outer paper. I will then cut the outer paper on the hole. This allows the sheetrock to act as its own joint tape. Works great on holes in walls from door knobs and such, but I've never tried it on an overhead repair since ever overhead repair that I've ever done is large enough that I needed to cut it out to the nearest ceiling joists anyway.

I seldom have luck trying to cut drywall to the center of a stud. The problem is that your blade ends up trying to cut through the old nails or screws. It's much easier and faster to not even try. Just cut flush to the edge of the stud - takes only a couple of minutes then to put in 1x4 or 2x4 blocking. Collinleon's tip is a good one - I use his method to patch small holes where blocking would be a hassle. But if you've got blocking or stud availble, I would definitely put a few screws and tape it. Use paper tape in preference to mesh for patch repairs... it's thinner than the mesh stuff and will feather in better. Personally I wouldn't worry about a great tight fit on the new panel. I would rather have 1/8 or so, to allow some joint compound to squeeze in there. Finally, I always use 20 minute setting compound for the first (tape) layer. It's a lot stronger and makes the job go faster. Regular compound spreads nicer and sands easier, so it's better for the finish coat.